Radiation and Heat Source Effects on MHD Free Convection Flow over an Inclined …article.ajoam.org/pdf/10.11648.j.ajam.20200804.14.pdf · on inclined magnetic field with dissipation

American Journal of Applied Mathematics 2020; 8(4): 190-206

http://www.sciencepublishinggroup.com/j/ajam

doi: 10.11648/j.ajam.20200804.14

ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online)

Radiation and Heat Source Effects on MHD Free Convection Flow over an Inclined Porous Plate in the Presence of Viscous Dissipation

Ekakitie Omamoke1, *

, Emeka Amos2, Kubugha Wilcox Bunonyo

3

1Department of Mathematics, Bayelsa Medical University, Yenagoa, Nigeria 2Department of Mathematics, Rivers State University, Port Harcourt, Nigeria 3Department of Mathematics and Statistics, Federal University Otuoke, Yenagoa, Nigeria

Email address:

*Corresponding author

To cite this article: Ekakitie Omamoke, Emeka Amos, Kubugha Wilcox Bunonyo. Radiation and Heat Source Effects on MHD Free Convection Flow over an

Inclined Porous Plate in the Presence of Viscous Dissipation. American Journal of Applied Mathematics. Vol. 8, No. 4, 2020, pp. 190-206.

doi: 10.11648/j.ajam.20200804.14

Received: June 18, 2020; Accepted: July 3, 2020; Published: July 17, 2020

Abstract: This study analyzes radiation, viscous dissipation and heat source effects on magneto-hydrodynamic free

convection flow, of a viscous incompressible fluid over an inclined porous plate. Applying the perturbation technique, the

solution of a set of ordinary differential equations are gotten as a result of reducing the non-linear partial differential equations

of motion, energy and diffusion, which is solved analytically for velocity, temperature and the concentration distribution. The

effect of Radiation, viscous dissipation and heat source on the velocity, temperature, concentration, skin friction, heat transfer

and rate of mass flux distribution is plotted graphically using Mathematica 12 software and discussed. It is observed that

increased Magnetic field reduces the velocity profile, increases the temperature, skin friction and heat transfer profile. Increase

in radiation reduces heat transfer causing a mixed effect on the velocity, temperature and skin friction while an increase in the

heat source causes a turbulent effect on the velocity, temperature and skin friction profile. Increase in porosity reduces the

velocity, temperature, skin friction and heat transfer profile and finally parameters such as Chemical reaction, Grashof

concentration number and Schmidt number had no effect on the velocity, temperature, skin friction and heat transfer profile.

Keywords: Free Convection, Radiation, Viscous Dissipation, Heat Source, Inclined Plate, Porous Plate,

Magneto Hydrodynamic (MHD)

1. Introduction

Magneto-hydrodynamic Free convection flow with

radiation, viscous dissipation and heat source effect on the

boundary layer control and fluid flow affects the

performances of engineering, science and medical devices

and systems. There are rich application in MHD power

generators, nuclear reactors, Human Blood flow and Water

flows etc. MHD free convective heat and mass transfer flow

for vertical, horizontal and inclined surfaces have attracted

many researchers because of its rich application in Medicine,

engineering and technology, agriculture, space technology

and many more. Amos and Ekakitie [1] studied MHD free

convective flow over an inclined porous surface with variable

suction and radiation effect while Isreal-Cookey et al. [2]

studied the influence of viscous dissipation and radiation on

unsteady MHD free-convection flow past an infinite heated

vertical plate in a porous medium with time-dependent

suction. Mebine [3] studied the thermal solutal MHD flow

with radiative heat transfer over oscillating plate in a fluid

that is chemically active and got the solutions for the

unsteady velocity, temperature and concentration with

Laplace transform on the assumption of an optically thin

medium, and linear differential approximation model for the

radiative flux. Venkateswarlu [4] analyzed the MHD

unsteady flow of viscous, incompressible electrical

American Journal of Applied Mathematics 2020; 8(4): 190-206 191

conduction fluid over a vertical porous plate under the

influence of thermal radiation and chemical reaction while

Sankar et al. [5] studied radiation and chemical reaction

effects on an unsteady MHD convection flow past a vertical

moving porous plate embedded in a porous medium with

viscous dissipation. Prasad et al. [6] did a study on the heat

and mass transfer for MHD flow of nano fluid with radiation

absorption while Rao et al. [7] did a study on Unsteady MHD

free convective double diffusive and dissipative visco-elastic

fluid flow in a porous medium with suction. Venkateswarlu et

al. [8] studied Soret, hall current, rotation, chemical reaction

and thermal radiation effects on unsteady MHD heat and

mass transfer of a natural convection flow past an accelerated

vertical plate, Reddy et al. [9] did a study on Soret and

Defour effects on radiation absorption fluid in the presence of

exponentially varying temperature and concentration in a

conducting field and Rojaa et al. [10] studied radiation and

chemical reaction effect on MHD free convective flow of a

micro polar fluid bounded by a vertical infinite surface with a

viscous dissipation and uniform or constant suction.

Venkateswarlu et al. [11] did a study on unsteady MHD flow

of a viscous fluid past a vertical porous plate under

oscillatory suction velocity while Ojemeri et al. [12] studied

the effect of Soret and radial magnetic field of a free

convective slip flow in a viscous reactive fluid towards a

vertical porous cylinder. Reddy et al. [13] studied radiation

and mass transfer effects on unsteady MHD free convection

flow of an incompressible viscous fluid past a moving

vertical cylinder and Rammana Reddy et al. [14] studied the

influence of chemical reaction, radiation and rotation on

MHD Nano fluid flow past a permeable flat surface in a

medium that is porous. Krupa et al. [15] studied the

numerical analysis on the effect of diffusion-thermo and

thermo-diffusion on two-phase boundary layer flow past a

stretching sheet with fluid particles suspension and chemical

reaction. Salawu and Dada [16] studied radiative heat

transfer of variable viscosity and thermal conductivity effects

on inclined magnetic field with dissipation in a non-Darcy

medium and Das [17] studied the effect of thermal radiation

on heat and mass transfer flow of MHD micro-polar fluid in

a rotary frame of reference. Ekakitie and Amos [18] studied

the impact of chemical reaction and heat source on MHD free

convection flow over an inclined porous surface. Hence in

this research we will be looking at the effect of radiation,

viscous dissipation and heat source on MHD free convective

flow over an inclined porous plate. Pertinent parameters such

as suction and radiation had an effect on velocity profile,

temperature profile, concentration profile and heat transfer.

2. Mathematical Formulation

We consider a free convective unsteady flow of a viscous

incompressible, chemically reactive, radiative and viscous

hydro magnetic fluid over an inclined plate at an angle α in

the vertical direction. A magnetic field of constant intensity

B₀ is applied in the y – direction. The plate moves uniformly

with velocity V in the x – direction, the temperature �� > �� and concentration �� > ��are conditions at the wall and

ambient regions respectively. We assume a homogeneous

chemical reaction of first and second order and that of

Boussinesq approximation is valid. We apply a constant

magnetic field of intensity B₀ in the y – direction.

Thus the flow equations are as follows.

�

�� = 0 (1)

��

�� + v� ��

�� = � ��

�� − ��

� + �� + !"#�� − ��$%&' ∝ + !) #�� − ��$%&' ∝ (2)

�"

�� + v� �"

�� = *�)+

��"

�� + ,�)+

��

��-

− *�)+

�./

�� + 0��)+

#�� − ��$ + ��

�)+��-

(3)

�)

�� + v� �)

�� = 1 ��)

�� − 23�#�� − ��$ (4)

The velocity (u), temperature (T) and concentration (C) fields in the above equations have the boundary conditions:

�� = 4; v� = −V71 + 9:;<�=; �� = �� ; �� = �� ; >? @ = 0 (5)

� = A#?$ = V�#1 + 9:;<�$; �� = �� ; �� = ��

� >' @ → ∞ (6)

Considering (3) by using the Rosseland diffusion approximation where the radiative heat flux q is given by

D3� = − E�

F��"

�� = − E�

F� ∇�� (7)

Where H� is the Stefan – Boltzmann constant and I� is the rosseland mean absorption coefficient. Assuming that the

temperature differences within the flow are sufficiently small that �� may be expressed as a linear function of temperature

�� = 4�KF� − 3�K

� This implies that

D3� = − MN�"O

P

F��"�� (8)

192 Ekakitie Omamoke et al.: Radiation and Heat Source Effects on MHD Free Convection Flow over an

Inclined Porous Plate in the Presence of Viscous Dissipation

In order to write the governing equations and boundary conditions in dimensionless form, the following non-dimensional

quantities are introduced

@ = Q�

� ; � = �

R ; ? = Q��

� ; v = S

Q ; T = "U"V"W U"V

; � = )U)V)W U)V

(9)

X- = Y��O�

�Q� ; N- = ��

*Q� ; [" = \��]^7"WU"V=RQ� ; [) = \��_)7)WU)V=

RQ� ; � = ,� ;

3̀ = ,)+* ; a- = MN�"O

P

* ; b) = R�

)+7"W U"V= ; cd = �e ; f = 0�Y

�)+Q� (10)

The momentum, energy and diffusion equation in dimensionless form is written as

�� − #1 + 9:;<�$ ��

�� = �� − g#X- + h-$�i + ["T%&' ∝ +[)�%&' ∝ (11)

3̀�^�� − 3̀#1 + 9:;<�$ �^

�� = #1 + a-$ ��^�� + 3̀b) j��

��k-

+ f`lT + X-`lb%� (12)

cd�)�� − cd#1 + 9:;<�$ �)

�� = ��)�� − cd23� (13)

The corresponding boundary conditions in non-

dimensional form are:

� = U = 0, T = 1, � = 1, >? @ = 0

� → 1 + 9:;<� , T → 0, � → 0, >? @ → ∞ (14)

In the above equations u and v are the velocity components,

x and y are the Cartesian coordinates, α is the angle of

inclination in the vertical direction of the semi-infinite

moving plate. ? is the time, �� is the temperature at the wall,

�� is the reference temperature and g is the acceleration due

to gravity. v is the suction velocity, !o is the constant

magnetic field, k is the porosity parameter, �p is the specific

heat capacity and X is the magnetic parameter. [" is Grashof

temperature number, 3̀ is the Prandtl number, Sc is Schmidt

number, R is the radiation parameter, D3� is the radiation heat

flux, Ec is the Eckert number, 9 is the small positive constant,

q is the density, r" is the coefficient of volume expansion

due to temperature and r) is the coefficient of volume

expansion due to concentration. h is the porosity number, s

is the free stream frequency oscillation,, ∂d is the electrical

conductivity, [d is the Grashof diffusion number, Kr is the

chemical reaction parameter, T is the temperature and �� is

the concentration at the wall.

3. Method of Solution

Equations (2) – (4) are coupled nonlinear partial

differential equations that is solved analytically and then

reduced to a set of ordinary differential equations with the

expressions for velocity (u), temperature ( T ) and

concentration (C) of the fluid in dimensionless form as

follows:

�#@, ?$ = �u#@$ + 9:;<��M + 0#9-$ (15)

T#@, ?$ = Tu#@$ + 9:;<�TM + 0#9-$ (16)

�#@, ?$ = �u#@$ + 9:;<��M + 0#9-$ (17)

Substituting equation (15) to (17) in the set of equations

(11), (12) and (13) and equating non – harmonic and

harmonic terms and neglecting the higher order terms of

0#9-$, the following set of ordinary differential equations are

obtained with their boundary conditions.

�ov v + �o

v − #N- + X-$�o = −["To%&' ∝ − [)�o%&' ∝ (18)

#1 + a-$Tov v + 3̀To

v + f`lTo = − 3̀bd�ov � − X-`lb%�o (19)

�ov v + cd�o

v − cd23�u = 0 (20)

�Mv v + �M

v − #N- + X- + ws$�M = −�ov − ["TM%&' ∝ − [)�M%&' ∝ (21)

#1 + a-$TMv v + 3̀TM

v + gf`l − 3̀wsiTM = − 3̀Tov − 2 3̀bd�o

v �Mv − X-`lb%�M (22)

�Mv v + cd�M

v − #cd23 + cdws$�M = −cd�ov (23)

Boundary conditions

�o = �; �M = 0; To = 1; , TM = 0; �o = 1; �M = 0 >? @ = 0 (24)

�o → 0; �M → 0; To → 0; , TM → 0; �o → 0; �M → 0 >' @ → ∞ (25)

For 0(9$ equations

To solve the nonlinear coupled equations (18) – (23) we assume that the dissipation parameter (Eckert number bd ≪ 1$ is


small, and therefore, advance an asymptotic expansion for the flow temperature and velocity as follows using the boundary

condition ins equation (24 – 25):

�o#@$ = �uM#@$ + bd�u-#@$

To#@$ = TuM#@$ + bdTu-#@$ (26)

�M#@$ = �MM#@$ + bd�M-#@$

TM#@$ = TMM#@$ + bdTM-#@$

Substituting Eq. (26) into Eq. (18) – (23) we obtain the following sequence of approximation

�oMv v + �oM

v − #N- + X-$�uM − ["ToM%&' ∝ − [)�o%&' ∝ (27)

#1 + a-$ToMv v + 3̀ToM

v + f`lToM = 0 (28)

�o-v v + �o-

v − #X- + N-$�u- = −["To-%&' ∝ (29)

#1 + a-$To-v v + 3̀To-

v + f`lTo- = − 3̀�oMv - − X-`l�oM (30)

Boundary conditions

�oM = �; ToM = 1; �o- = 0 = To- >? @ = 0 (31)

�oM → 0; ToM → 0; �o- → 0; To- → 0; >' @ → ∞ (32)

�MMv v + �MM

v − #N- + X- + ws$�MM = −�oMv − ["TMM%&' ∝ − [)�M%&' ∝ (33)

#1 + a-$TMMv v + `lTMM

v + gf`l − `lwsiTMM = −`lToMv (34)

�M-v v + �M-

v − #N- + X- + ws$�M- = −�o-v − ["TM-%&' ∝ (35)

#1 + a-$TM-v v + `lTM-

v + gf`l − `lwsiTM- = −`lTo-v − 2`l�MM

v �oMv − X-`l�MM (36)

Boundary condition

�MM = 0; TMM = 0; �M- = 0 = TM-; >? @ = 0 (37)

�MM → 0; TMM → 0; �M- → 0; TM- → 0; >' @ → ∞ (38)

For O (Ec) equations solving equations (27) – (30) with the boundary conditions (30) – (31) and equation (32) – (35)

satisfying the boundary condition (37) – (38) then equation (25) becomes

�o#@$ = zM:U{P�−z-:U{�� − zF:U{|� − 1 + bd}zME:U{~� − zM�:U{�� + zMN:U-{P� + zM�:U-{�� + zM�:U-{|� −zM�:U#{��{P$� − z-o:U#{|�{P$� + z-M:U#{|�{�$� + z--:U{P� − z-F:U{�� + z-E:U{|�� (39)

To#@$ = :U{�� + bd}zE:U{�� − z�:U-{P� − zN:U-{�� − z�:U-{|� + z�:U#{��{P$� + z�:U#{|�{P$� − zMo:U#{|�{�$� −zMM:U{P� + zM-:U{�� + zMF:U{|�� (40)

�M#@$ = z-�:U{�� − zFo:U{�� − zF-:U{�� − zFE:U{P� − #zFM + zF�$:U{�� − #zFF + zFN$:U{|� + bd}zNo:U{|�� −zNM:U{�� − zN-:U{�� + zNF:U{P� − zNE:U{�� − zN�:U{|� + zNo:U-{P� + zNM:U-{�� + zN-:U-{|� − zNF:U#{��{P$� −

zNE:U#{|�{P$� − zN�:U#{|�{�$�+zNN:U{�� − zN�:U{�� − zN�:U{�� − zN�:U{~� + z�o:U-{P� + z�M:U-{�� + z�-:U-{|� −z�F:U#{��{P$� − z�E:U#{|�{P$� + z��:U#{|�{�$� + z�N:U#{P�{�$� − −z��:U#{P�{�$�U − z��:U#{P�{�$� + z��:U#{��{�$� +

z�o:U#{��{�$� + z�M:U#{��{�$� − z�-:U#{|�{�$� + z�F:U#{|�{�$� + z�E:U#{|�{�$�� (41)

TM#@$ = z-�:U{�� + z-�:U{�� + bd}zF�:U{�� + zF�:U{�� − zF�:U-{P� − zEo:U-{�� − zEM:U-{|� + zE-:U#{��{P$� +zEF:U#{|�{P$� − zEE:U#{|�{�$� − zE�:U{P�+zEN:U{��+zE�:U{|� − zE�:U#{P�{�$� + zE�:U#{P�{�$� +z�o:U#{P�{�$� − z�M:U#{��{�$� − z�-:U#{��{�$� − z�F:U#{��{�$� + z�E:U#{|�{�$� − z��:U#{|�{�$� −

z�N:U#{|�{�$� − z��:U{�� + z��:U{�� + z��:U{�� (42)

Substituting equation (39) – (42) into equation (15) – (16) we obtain the velocity, temperature and concentration profiles

respectively as

�#@, ?$ = zM:U{P�−z-:U{��−zF:U{|� + bd}zME:U{~� − zM�:U{�� + zMN:U-{P� + zM�:U-{�� + zM�:U-{|� −



zM�:U#{��{P$� − z-o:U#{|�{P$� + z-M:U#{|�{�$� + z--:U{P� − z-F:U{�� + z-E:U{|�� + 9:;<��z-�:U{�� −zFo:U{�� − zF-:U{�� − zFE:U{P� − #zFM + zF�$:U{�� − #zFF + zFN$:U{|� + bd}zNo:U{|�� − zNM:U{�� − zN-:U{�� +

zNF:U{P� − zNE:U{�� − zN�:U{|� + zNo:U-{P� + zNM:U-{�� + zN-:U-{|� − zNF:U#{��{P$� − zNE:U#{|�{P$� −zN�:U#{|�{�$�+zNN:U{�� − zN�:U{�� − zN�:U{�� − zN�:U{~� + z�o:U-{P� + z�M:U-{�� + z�-:U-{|� −

z�F:U#{��{P$� − z�E:U#{|�{P$� + z��:U#{|�{�$� + z�N:U#{P�{�$� − −z��:U#{P�{�$�U − z��:U#{P�{�$� +z��:U#{��{�$� + z�o:U#{��{�$� + z�M:U#{��{�$� − z�-:U#{|�{�$� + z�F:U#{|�{�$� + z�E:U#{|�{�$�� (43)

T#@, ?$ =:U{�� + bd}zE:U{�� − z�:U-{P� − zN:U-{�� − z�:U-{|� + z�:U#{��{P$� + z�:U#{|�{P$� − zMo:U#{|�{�$� −zMM:U{P� + zM-:U{�� + zMF:U{|�� + 9:;<��z-�:U{�� + z-�:U{�� + bd}zF�:U{�� + zF�:U{�� − zF�:U-{P� −

zEo:U-{�� − zEM:U-{|� + zE-:U#{��{P$� + zEF:U#{|�{P$� − zEE:U#{|�{�$� − zE�:U{P�+zEN:U{��+zE�:U{|� −zE�:U#{P�{�$� + zE�:U#{P�{�$� + z�o:U#{P�{�$� − z�M:U#{��{�$� − z�-:U#{��{�$� − z�F:U#{��{�$� +z�E:U#{|�{�$� − z��:U#{|�{�$� − z�N:U#{|�{�$� − z��:U{�� + z��:U{�� + z��:U{�� (44)

�#@, ?$ = :U{|� + 9:;<��z-�:U{�� + z-N:U{|�� (45)

The physical quantities of interest are the wall shear stress �� is given by

�� = � ��

�@� = q�o-��#0$

The local skin friction factor

�� = �W�Y��

= ��#0$ = −�FzM + �-z- + �MzF + bdg−��zME + �EzM� − 2�FzMN − 2�-zM� − 2�MzM� + #�- + �F$zM� +#�M + �F$z-o − #�M + �-$z-M − �Fz-- + �-z-F + �Mz-Ei + 9g−��z-� + ��zFo + �-#zFM + zF�$ + �NzF- +

�MzFF − �FzFE + �-zF� + �MzFN + bd#−�MozNo + ��zNM + �EzN- − �FzNF + �-zNE + �MzN� − ��zNN + ��zN� +�NzN� + ��zN� − 2�Fz�o − 2�-z�M − 2�Mz�- + #�- + �F$z�F + #�M + �F$z�E − #�M + �-$z�� − #�F + ��$z�N +

#�F + ��$z�� + #�F + �N$z�� − #�- + ��$z�� − #�- + ��$z�o − #�- + �N$z�M + #�M + ��$z�- − #�M + ��$z�F −#�M + �N$z�Ei (46)

The Local Surface heat Flux D3 = UE��"�

F�� where I is the effective thermal conductivity

Local Nusselt Number h�� = ./"WU"V

where D3 = UMN��"�PS��#"WU"V$F��Y

�^��

Then h�� = UMN��"�PS��

F��Y�^�� where a�� = UMN��"�PS��

F��Y

��

= �^�� = T�#0$ = −�- + bdg−�EzE + 2�Fz� + 2�-zN + 2�Mz� − #�- + �F$z� = −#�M + �F$z� + #�M +

�-$zMo + �FzMM − �-zM- − �MzMFi + 9g−��z-� − �-z-� + bd#−��zF� − �EzF� + 2�FzF� + 2�-zEo + 2�MzEM −#�- + �F$zE- − #�M + �F$zEF + #�M + �-$zEE + �FzE� − �-zEN − �MzE� + #�F + ��$zE� − #�F + ��$zE� −

#�F + �N$z�o + #�- + ��$z�M + #�- + ��$z�- + #�- + �N$z�F − #�M + ��$z�E + #�M + ��$z�� + #�M + �N$z�N +��z�� − ��z�� − �Nz��$i (47)

The local surface mass flux is given by

��

= − �)�� = ��#0$ = −�M − 9#�Nz-� + �Mz-N$ (48)

4. Results and Discussions

The effect of various parameters on the velocity,

temperature, concentration, skin friction, heat transfer and

mass flux rate is discussed in this session. An increase in

magnetic field reduced the velocity profile in Figure 1 and

then increased the temperature profile in Figure 11. This

increases the skin friction and increases the heat transfer in

Figure 20 and Figure 25. The reduction of the velocity flow

was due to Lorentz force which thickens the boundary layer

and causes increase in skin friction and decrease in the heat

transfer.

An increase in radiation causes a decrease in the velocity

and temperature profile in Figure 2 and Figure 9. This in turn

reduces both the skin friction in Figure 19 and the heat

transfer profile in Figure 24.

An increase in heat source had a mixed effect on the

velocity profile in Figure 3, temperature profile decreases

near the plate in Figure 10 and skin friction profile reduces in

Figure 18 which decreases the heat transfer profile in Figure

23.

In Figure 4, Figure 12, Figure 21 and Figure 26, an

increase in porosity reduces the velocity, temperature, skin

friction and heat transfer profile. There was an opposite


effect in Figure 6, Figure 15, Figure 22 and Figure 27 where

an increase in viscous dissipation increases the velocity,

temperature, skin friction and heat transfer profile.

In Figure 5 and Figure 13, an increase in Grashof number

increases the velocity and temperature profile while an

increase in Prandtl number in Figure 8 increases the velocity

profile. In Figure 7 and Figure 14, an increase in the angle of

inclination causes a reduction in velocity and temperature

profile near the plate but an increase in velocity and

temperature profile at the boundary layer.

In Figure 16 an increase in chemical reaction reduces the

concentration profile while an increase in Schmidt number

reduces the concentration profile in Figure 17, but increases

the mass transfer in figure 28. Figures 29, 30 and 31 shows

that there was no effect of Chemical reaction (Kr), Schmidt

number (Sc) and Grashof diffusion number (Gc) on the

velocity profile.

Figure 1. Velocity Profile with Variation of Magnetic Field Parameter X for f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 2. Velocity Profile with Variation of Radiation Parameter a for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, h �0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 3. Velocity Profile with Variation of Heat Source Parameter for H for X � 2, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 4. Velocity Profile with Variation of Porosity Parameter h for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, £ � 10, s � 1, ? � 1, 9 � 1.



Figure 5. Velocity Profile with Variation of Grashof Temperature Parameter [3 for X � 2, f � 0.5, 3̀ � 0.71, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 6. Velocity Profile with Variation of Viscous Dissipation Parameter Ec for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 7. Velocity Profile with Variation of angle of inclination Parameter £ for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% �0.01, a � 0.1, h � 0.1, s � 1, ? � 1, 9 � 1.

Figure 8. Velocity Profile with Variation of Prandtl Number Parameter Pr for X � 2, f � 0.5, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a � 0.1, h �0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 9. Temperature Profile with Variation of Radiation Parameter R for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, h �0.1, £ � 10, s � 1, ? � 1, 9 � 1.


Figure 10. Temperature Profile with Variation of Heat Source Parameter f for X � 2, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 11. Temperature Profile with Variation of Magnetic Field Parameter X for f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 12. Temperature Profile with Variation of Porosity Parameter N for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 13. Temperature Profile with Variation of Grashof Temperature Parameter [l for X � 2, f � 0.5, 3̀ � 0.71, [d � 10, cd � 0.22, 2l � 2, b% �0.01, a � 0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 14. Temperature Profile with Variation of angle of inclination Parameter £ for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% �0.01, a � 0.1, h � 0.1, s � 1, ? � 1, 9 � 1.



Figure 15. Temperature Profile with Variation of Viscous Dissipation Parameter b% for X � 2, c � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 16. Concentration Profile with Variation of Chemical Reaction Parameter 2l for c% � 0.22, s � 1, ? � 1, 9 � 1.

Figure 17. Concentration Profile with Variation of Schmidt number Parameter c% for 2l � 2, s � 1, ? � 1, 9 � 1.

Figure 18. Skin Friction Profile with Variation of Heat Source Parameter H for X � 2, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 19. Skin Friction Profile with Variation of Radiation Parameter a for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% �0.01, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.


Figure 20. Skin Friction Profile with Variation of Magnetic Field Parameter X for f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 21. Skin Friction Profile with Variation of Porosity Parameter h for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 22. Skin Friction Profile with Variation of Viscous Dissipation Parameter b% for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l �2, a � 0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 23. Heat Transfer Profile with Variation of Heat Source Parameter for X � 2, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 24. Heat Transfer Profile with Variation of Radiation Parameter for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, h �0.1, £ � 10, s � 1, ? � 1, 9 � 1.



Figure 25. Heat Transfer Profile with Variation of Magnetic Field Parameter for f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 26. Heat Transfer Profile with Variation of Porosity Parameter for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l � 2, b% � 0.01, a �0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 27. Heat Transfer Profile with Variation of Viscous Dissipation Parameter b% for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, 2l �2, a � 0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 28. Mass Transfer Profile with Variation of Chemical Reaction Parameter 2l for c% � 0.22, s � 1, ? � 1, 9 � 1.

Figure 29. Velocity Profile with Variation of Chemical Reaction Parameter 2l for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, cd � 0.22, b% � 0.01, a �0.1, h � 0.1, £ � 10, s � 1, ? � 1, 9 � 1.


Figure 30. Velocity Profile with Variation of Schmidt Number Parameter c% for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, , b% � 0.01, a � 0.1, h �0.1, £ � 10, s � 1, ? � 1, 9 � 1.

Figure 31. Velocity Profile with Variation of Grashof diffusion Parameter ¤¥ for X � 2, f � 0.5, 3̀ � 0.71, [3 � 15, [d � 10, , b% � 0.01, a � 0.1, h �0.1, £ � 10, s � 1, ? � 1, 9 � 1.

5. Conclusion

Radiation, Viscous dissipation and heat source effects on

MHD free convection flow over an inclined porous surface

are summarized as follows,

a) Increased Magnetic field reduces the velocity profile,

increases the temperature, decreases skin friction and

increases heat transfer profile.

b) Increase in radiation reduces the velocity, temperature,

Skin friction and heat transfer profile.

c) An increase in the heat source caused a mixed effect on

the velocity, decreases the temperature near the plate,

decreases skin friction profile and heat transfer profile.

d) Increase in porosity reduces the velocity, temperature,

Skin friction and heat transfer profile.

e) Increase in the angle of inclination of the porous plate

initially reduces the velocity and temperature profile but

later increases the velocity and temperature profile,

hence having a mixed effect on both velocity and

temperature.

f) Increase in viscous dissipation increases velocity,

temperature, Skin friction and heat transfer profile.

g) From the research so far parameters such Grashof

concentration number (Gc), Schmidt number (Sc) and

Chemical reaction (Kr) had no effect on the velocity

profile and temperature profile.

Appendix

X- � ��-HdfK��

q�u�� ; N- � �-

2�u�� ; [3 � �!"T#��

� � ��$AK�u�

� ; [d � �!)�#�� $

AK�u�� ; 3̀ � ��p

I ; a- � 16�H��KF

3I� ;

b) � SO�

)+7"W U"V= ; cd � �e ; �M � ��√7��E��*/=

- ; �- � U /̈U√j /̈��E©¨37M��=k-#M��$ � �E;

�F � 1 � √#1- � 4#N- � X-$$2 � ��; �N �

cd � √jcd- � 4#cd23 � cdws$k

2

�� /̈�√j /̈�UE#©U /̈;<$7M��=k-#M��$ � ��; �� M�√7M��E#ª��«��;<$=

- � �Mo

A1 � A2 � A3;

A2 � GrCosg£im2- � m2 � #X- � h-$ ;

A3 � GcCosg£im2- � m2 � #X- � h-$ ;



A4 = A5 + A6 + A7 − A8 − A9 + A10 + A11 − A12 − A13;

A5 = Prm3-A1-

4#1 + a-$m3- − 2Prm3 + fPr ;

A6 = Prm2-A2-

4#1 + a-$m2- − 2Prm2 + fPr ;

A7 = Prm1-A3-

4#1 + a-$m1- − 2Prm1 + fPr ;

A8 = 2Prm2m3A1A2#1 + a-$#m2 + m3$- − Pr#m2 + m3$ + fPr ;



A11 = Prm3-A1-

#1 + a-$m3- − Prm3 + fPr ;

A12 = Prm2-A2-

#1 + a-$m2- − Prm2 + fPr ;

A13 = Prm1-A3-

#1 + a-$m1- − Prm1 + fPr ;

A15 = GrA4Cosg£im4- − m4 + #X- + h-$ ;

A16 = GrA5Cosg£i4m3- + 2m3 − #X- + h-$ ;

A17 = GrA6Cosg£i4m2- + 2m2 − #X- + h-$ ;

A18 = GrA7Cosg£i4m1- + 2m1 − #X- + h-$ ;

A19 = GrA8Cosg£i#m2 + m3$- + #m2 + m3$ − #X- + h-$ ;

A20 = GrA9Cosg£i#m1 + m3$- + #m1 + m3$ − #X- + h-$ ;

A21 = GrA10Cosg£i#m1 + m2$- + #m1 + m2$ − #X- + h-$

A22 = GrA11Cosg£im3- + m3 − #X- + h-$

A23 = GrA12Cosg£im2- + 2m2 − #X- + h-$

A24 = GrA13Cosg£i4m1- + 2m1 − #X- + h-$

A25 = −A26


A26 = m1Scm1- − Scm1 − #ScKr + wsSc$

A27 = −A28

A28 = Prm2#1 + a-$m2- − Prm2 + #fPr − Prws$

A29 = A30 + A31 + A32 + A33 − A34 + A35 + A36

A30 = GrA27Cosg£im7- − m7 − #X- + h- + ws$

A31 = GrA28Cosg£im2- − m2 − #X- + h- + ws$

A32 = GcA25Cosg£im6- − m6 − #X- + h- + ws$

A33 = GcA26Cosg£im1- − m1 − #X- + h- + ws$

A34 = m3A1m3- − m3 − #X- + h- + ws$

A35 = m2A2m2- − m2 − #X- + h- + ws$

A36 = m1A3m1- − m1 − #X- + h- + ws$

A37 = 1

A38 = Prm4A4#1 + a-$m4- − Prm4 + #fPr − Prws$

A39 = #2Prm3$#m3A1A34 + A5$4#1 + a-$m3- − Prm3 + #fPr − Prws$

A40 = #2Prm2$#m2A31A2 + m2A2A35 + A6$4#1 + a-$m2- − Prm2 + #fPr − Prws$

A41 = #2Prm1$#m1A3A33 + m1A3A36 + A7$4#1 + a-$m1- − Prm1 + #fPr − Prws$

A42 = Pr#m2 + m3$A8 + #2m2m3$#A1A31 + A1A35 + A2A34$4#1 + a-$#m2 + m3$- − Pr#m2 + m3$ + #fPr − Prws$

A43 = Pr#m1 + m3$A9 + #2m1m3$#A1A33 + A1A36 + A3A34$4#1 + a-$#m1 + m3$- − Pr#m1 + m3$ + #fPr − Prws$

A44 = Pr#m1 + m2$A10 + #2m1m2$#A2A33 + A2A36 + A3A31 + A3A35$4#1 + a-$#m1 + m2$- − Pr#m1 + m2$ + #fPr − Prws$

A45 = Pr#m3A11 + X-A34$#1 + a-$m3- − Prm3 + #fPr − Prws$

A46 = Pr#m2A12 + X-#A31 + A35$$#1 + a-$m3- − Prm3 + #fPr − Prws$

A47 = Pr#m1A13 + X-#A33 + A36$$#1 + a-$m1- − Prm1 + #fPr − Prws$



A48 = 2Prm3m8A1A29#1 + a-$#m3 + m8$- − Pr#m3 + m8$ + #fPr − Prws$









A57 = X-PrA29#1 + a-$m1- − Prm1 + #fPr − Prws$



A60 = A61 + A62 − A63 + A64 + A65 − A66 + A67 + A68 + A69 − A70 − A71 − A72 + A73 + A74 − A75 − A76 + A77 + A78− A79 − A80 − A81 + A82 − A83 − A84

A61 = GrA37Cosg£im9- − m9 + #X- + h- + ws$

A62 = GrA38Cosg£i + #m4A15$m4- − m4 + #X- + h- + ws$







A69 = m5A14m5- − m5 + #X- + h- + ws$


A70 = GrA39Cosg£i + #2m3A16$m3- − m3 + #X- + h- + ws$



A73 = GrA42Cosg£i + #m2 + m3$A19#m2 + m3$- − #m2 + m3$ + #X- + h- + ws$



A76 = GrA48Cosg£i#m3 + m8$- − #m3 + m8$ + #X- + h- + ws$









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